General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

epresent them as a data type, where the constructors conform to the three basic shapes they are
in. So a circle of radius r would be represented as the constructor term Circle $r$ (what else).
Data Types Example (Geometric Shapes)
describe three kinds of geometrical forms as mathematical objects
r
Circle (r)
a
Square (a)
Mathematically: R + ⊎ R + ⊎ ((R + × R + × R + ))
In SML: approximate R + by the built-in type real.
datatype shape =
Circle of real
| Square of real
| Triangle of real * real * real
This gives us the constructor functions
Circle : real -> shape
Square : real -> shape
Triangle : real * real * real -> shape
Some experiments:
- Circle 4.0
Circle 4.0 : shape
- Square 3.0
Square 3.0 : shape
- Triangle(4.0, 3.0, 5.0)
Triangle(4.0, 3.0, 5.0) : shape
c○: Michael Kohlhase 83
Data Types Example (Areas of Shapes)
a procedure that computes the area of a shape:
a
c
b
Triangle (a, b, c)
- fun area (Circle r) = Math.pi*r*r
| area (Square a) = a*a
| area (Triangle(a,b,c)) = let val s = (a+b+c)/2.0
in Math.sqrt(s*(s-a)*(s-b)*(s-c))
end
val area : shape -> real
New Construct: Standard structure Math (see [SML10])
some experiments
- area (Square 3.0)
9.0 : real
- area (Triangle(6.0, 6.0, Math.sqrt 72.0))
18.0 : real
c○: Michael Kohlhase 84
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